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I.I.T., Kharagpur
P-207
Kirchhoff Pre-Stack Depth Migration: effective tool for depth imaging
Rahul Jain*, IIT Kharagpur
Summary
Prestack Depth Migration (PSDM) is one of the most reliable seismic techniques for imaging subsurface structures because
of its ability to focus and position reflections in areas with strong lateral velocity variations. The comparison between
velocity sections obtained initially by velocity analysis in time domain on CMP gathers and the same obtained by horizon
velocity analysis shows the latter to be more coherent and geologically meaningful. Techniques based on model ray tracing,
such as coherency inversion allows more precise estimates of interval velocity to produce superior velocity models for
PSDM. This paper is an attempt to study the methodology and application of Kirchoff PSDM through model based interval
velocity estimation using coherency inversion technique followed by interval velocity depth model refinement using horizon
based tomography and subsequent Pre-Stack Depth Migration. The CMP gathers of an arbitrary 2D seismic line and the
Pre-Stack time migrated gathers generated from it were first analysed and by applying the above technique, PSDM sections
showed considerable improvement in imaging the subsurface picture in this particular case suggesting that in structurally
complex areas, this methodology can be suitably applied to derive much better geological result.
Introduction
Migration is a seismic data processing technique to map
seismic events onto their appropriate positions (Sheriff &
Geldart, 1995). Migration is done either in time domain or
depth domain depending on the complexity of lithology.
Time migration yields an inaccurate image in the presence
of strong lateral velocity variation associated with complex
overburden structure. In such a case, earth imaging is done
by depth migration. Strong lateral velocity variation causes
significant ray bending at layer boundaries, it gives rise to
non-hyperbolic behaviour of reflection times on CMP
gathers. As a result, amplitudes and travel times associated
with the reflection events with non-hyperbolic moveout are
distorted during conventional CMP stacking which is based
on the hyperbolic moveout assumption. This causes CMP
stack to depart from an ideal zero offset wave field.
Therefore, when depth migration is needed, in principle, it
is done before stack and not after stack (Yilmaz, 2001).
The first step in depth migration is to choose an interval
velocity depth model. The quality of the depth image
depends heavily on the input data, the inversion algorithm,
and a chosen class of models (number of reflection
interfaces, parameterization for interfaces, geometry and
velocities within the layers etc). Both Time and Depth
migration use a diffraction term for collapsing energy along
a diffraction hyperbola to its apex, only the depth migration
algorithms implement the additional thin-lens term that
explicitly account for lateral velocity variation. The general
workflow for pre stack depth migration (Furniss, 2000) is
as given below:
• Stacking velocity analysis along time horizons
• RMS velocity analysis along time migrated
horizons
• Stacking velocity refinement along time horizons
• RMS velocity refinement along time migrated
horizons
• Interval velocity and depth model creation
(coherency inversion)
• Interval velocity and depth model refinement and
modelling (tomography)
Velocity Estimation
The simplest method for estimating layer velocities is Dix
conversion of RMS Velocities (Dix, 1995). Dix Equation is
based on the assumptions that the layer boundaries are flat
and the offset range in estimating RMS velocities
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Kirchhoff Pre-Stack Depth Migration: effective tool for
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corresponds to small spread. Additionally, RMS Velocities
used in the equation are based on straight ray assumptions
thus ray bending at layer boundaries are not accounted for.
Thus for a layered earth model with dipping layer
boundaries and layer velocities with vertical and lateral
variations, more accurate methods are required to estimate
interval velocity such as Stacking velocity inversion or
Coherency Inversion.
Coherency Inversion
Coherency inversion is a process to identify the interval
velocity of a layer by using the propagation of ray tracing
along the model of time gather space. Velocity estimation
by coherency inversion provides a velocity-depth macro
model of the subsurface (Landa et al., 1988). The accuracy
for this model is critical for imaging technique such as
depth migration and inversion. At a given CMP, for a set of
interval velocities, travel times are computed over a range
of offsets based on ray tracing and compared with the
actual travel time curve observed on the CMP gather. Best
fitting curve as shown in Figure 1 identifies the proper
interval velocity at a given CMP. A velocity depth model
estimation using coherency inversion is conducted layer by
layer starting from the surface. The interval velocity profile
for the first layer H1 estimated from the Dix conversion is
first adopted and then the application of coherency
inversion is started with layer H2. Assuming that the
velocity depth model for the first (n-1) layers already has
been estimated, then for the nth layer, following are the
steps to do coherency inversion:
1. Perform normal incidence travel time inversion to
convert the time horizon corresponding to the base
layer boundary to a trial depth horizon using a
trial constant velocity assigned to the nth layer.
2. Compute CMP travel times of the nth layer for
specific analysis location using the known
overburden velocity depth model. The ray tracing is
used to compute the travel times to account for ray
bending at layer boundaries and incorporated vertical
gradient within the layer above.
3. Semblance is then computed at each CMP location to
measure the correlation between the recording
CMP gathers and model travel times curves for each
trial interval velocity value.
4. Pick the constant trial velocity as the nth layer velocity
for which the semblance is maximum. The maximum
semblance values describe the speed that makes the
CMP gathers flat.
Figure 1 Principle of coherency inversion
Model refinement through Tomography
Tomography is based on the principle that if migration is
carried out with correct velocity depth model, the image
gathers should be flat i.e. event depth is same at all receiver
(Tian-wen Lo et.al, 1994). It attempts to correct errors in
the velocity depth model by analysing the residual delays
after PSDM. Tomography of depth migrated gathers is a
method for refining the velocity-depth model. When pre-
stack depth migration is performed with an initial incorrect
velocity model derived from inversion methods based on
non-global approaches, the depth gathers will exhibit non-
flatness. The degree of non-flatness is a measurement of the
error in the model. Tomography uses this measurement of
non-flatness (residual moveout) as input and attempts to
find an alternative model, which will minimize the errors.
The tomographic principle attributes an error in time to an
error both in velocity and depth.
Tomography principle
Reflection travel time tomography is based on perturbing
the initial model parameters by a small amount and then
matching the change in travel times to the travel time
measurements made from residual velocity analysis of
image gathers (Sherwood et al., 1986; Kosloff et al., 1996).
In the usual implementation of reflection travel time
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Kirchhoff Pre-Stack Depth Migration: effective tool for
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tomography, the model parameters are perturbed while
preserving the offset value of the seismic data. The
tomographic update ∆p to the model parameters that
comprise the changes in the slowness and depths to layer
boundaries is given by the generalized linear inversion
(GLI) solutions (Eq. 1),
∆p= (LT L)-1 LT ∆t ………………… (1)
Where ∆t denotes the column vector that represents the
residual velocity move out times measured from the image
gathers, L is a sparse matrix – Its elements are in terms of
the slowness and depth parameters associated with the
initial model and T denotes the matrix transposition.
Representation of the subsurface
An important issue which needs to be addressed before
carrying out velocity analysis is the representation of the
subsurface. In the model based approach, the
subsurface is represented by formations which are
separated by interfaces. Within each formation, the
velocity is defined by a single valued velocity function of
the form (Eq. 2)
V (x,y,z) = v0(x,y) + g(x,y) * z ……………(2)
Where x and y represent the horizontal coordinates, and z is
the depth. v0(x,y) and g(x,y) are the background velocity
and the vertical gradient respectively. The model includes a
representation of the reflecting surfaces. One type of model
uses a layer-cake type formation sequence, where the depth
of each interface is given by a function of the horizontal
coordinates. A second approach uses a solid model with
closed triangulated surfaces which is able to handle
complex non layer-cake type structures. In the grid based
approach, the subsurface is represented by a three
dimensional velocity grid, where each grid point has an
assigned velocity. For each type of subsurface
representation there is a corresponding velocity
determination technique. As for global methods; model
based tomography updates a subsurface model, and
grid based tomography updates a velocity volume. Due to
non-uniqueness, the two approaches can yield quite
different results. Model based approaches are based on
generating a model from seismic interpretations and
thus include geological information. Therefore, whenever
possible, model based approaches should be preferred.
However, there are situations with low quality data when it
is difficult to define a model. Then the grid based
approach is more applicable.
There are situations where it is best to carry out part of the
analysis with a model based approach and another part with
a grid based approach. For example, in many cases it
is possible to build a model for the upper formations, but
the lower reflectors are not clearly interpretable across the
section. In such a situation it may be appropriate to use a
model based approach for the upper formations first,
and use a grid based approach for the deeper formations
afterwards. A typical update grid size is 50 CMP spacing’s
in the horizontal direction, and 200m in the vertical
direction. Within each grid cell, the slowness updates are
interpolated to the original velocity section sampling
increments by bilinear interpolation. This interpolation
assures smooth updates and avoids over parameterization
beyond the resolution of the seismic data.
Case study
CMP gather data along a 2D seismic line with their RMS
Velocity Model was used to perform Pre-Stack time
Migration to get a preliminary subsurface image (Figure 2)
and RMS velocity (Figure 3). To get more accurate image,
PSDM was performed.
Dix conversion is valid only for horizontally layered earth
models with constant layer velocities and small offsets,
which is not the case here so coherency inversion was
performed to get true interval velocities for each horizon. A
total of 7 Horizons were picked on PSTM section with the
help of seismic marker. For each horizon (H1-H7),
coherency inversion on CMP gather was performed to get
Horizon layer velocity. Coherency Inversion for horizon 2
is shown in Figure 4, where we can see the CMP raypath
tracing at a particular analysis location (yellow colour),
their corresponding CMP gathers and horizon-velocity
semblance spectrum (maximum in red colour).
Interval velocity section was generated with the help of
coherency inversion for the area shown in Figure 5. Now
Prestack depth migration using Kirchhoff’s algorithm was
carried out with the below depth interval velocity model
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Kirchhoff Pre-Stack Depth Migration: effective tool for
depth imaging
and time gathers as input. Here Quality control is required
whether our depth migrated gathers are flat or not after
PSDM. The output depth gathers were stacked separately
after proper mute as shown in Figure 6. The depth migrated
gathers were observed to be still not perfectly flat at larger
offsets, so further refinement in interval velocity model was
needed. To increase the reflection flatness in depth gather,
residuals were picked for each of the horizons and the
depth interval velocity model was updated using horizon
based tomography and final interval velocity was
generated. With this updated interval velocity and depth
model, PSDM was carried out. The resulting depth gathers
were found to be flat and the corresponding depth section
shown in Figure 7, showed improved structural features
with better standouts of the events in the zone of interest.
Figure 2 PSTM Stack data along a 2D seismic line
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Kirchhoff Pre-Stack Depth Migration: effective tool for
depth imaging
Figure 3 RMS Velocity Model
Figure 4 Coherency Inversion for Horizon 2 using GeoDepth Software
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Kirchhoff Pre-Stack Depth Migration: effective tool for
depth imaging
Figure 5 Interval Velocity Model generated by coherency Inversion
Figure 6: Prestack depth migrated stacked section and interval velocity section after coherency inversion
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Kirchhoff Pre-Stack Depth Migration: effective tool for
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Conclusions & Recommendations
1. Pre Stack Time Migration can effectively image
subsurface features when geology is not complex
and there are no lateral velocity changes.
2. Pre Stack Depth Migration is the most efficient
method to image subsurface when there is
structural complexity as well as lateral velocity
variation.
3. For building a depth interval velocity model, the ray
tracing technique instead of usual conventional
Dix’s approach should invariably be used.
4. To take this work further, an attempt can be made to
find a better algorithm than coherency
inversion which requires less labour and gives more
accuracy in the interval velocity model. Velocity
estimation with null space (Clapp, 1998) or
interval velocity estimation using edge preserving
regularization (Valenciano, 2004) may be helpful to
start with a new algorithm.
Acknowledgement
This paper is the outcome of the author’s MSc dissertation
work carried out at SPIC, ONGC Ltd, Mumbai on a 2D
seismic line. I take this opportunity to pay my gratitude to
my supervisors Prof. S.K Nath (former Head, Department
of Geology and Geophysics, I.I.T., Kharagpur) and Mr. A.
Ghosh & Mr. Rajesh Madan (SPIC) for their full and
continual support throughout the project. I am thankful to
Mr. D. Chatterjee (DGM (GP), SPIC) and Prof. Biswajit
Mishra, Head of the Department of geology and
geophysics, I.I.T., Kharagpur for providing me all possible
facilities for the work. I would also like to express my
heartfelt thanks to Mr. A.K. Bhartee (Chief Geophysicist,
SPIC) for many helpful discussions and for creating a very
friendly atmosphere at SPIC which led to the successful
completion of the work.
Figure 7 Final Depth Stacked Section after horizon based tomography
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Kirchhoff Pre-Stack Depth Migration: effective tool for
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References
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Furniss A. (2000), An integral Pre-stack depth migration
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Tian-wen Lo and Philip Inder Weisen (1994), Fundamentals
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